Constructing Sequences of Divided Powers

Embedding the Flag Representation in Divided Powers

A generalization of a theorem of Crabb and Hubbuck concerning the embedding of flag representations in divided powers is given, working over an arbitrary finite field F, using the category of functors from finite-dimensional F-vector spaces to F-vector spaces.

Constructing the Primitive Roots of Prime Powers

We use only addition and multiplication to construct the primitive roots of p from the primitive roots of p, where p is an odd prime and k ≥ 2.

Powers in Sturmian sequences

Initial Powers of Sturmian Sequences

We investigate powers of prefixes in Sturmian sequences. We obtain an explicit formula for ice(ω), the initial critical exponent of a Sturmian sequence ω, defined as the supremum of all real numbers p > 0 for which there exist arbitrary long prefixes of ω of the form u, in terms of its S-adic representation. This formula is based on Ostrowski’s numeration system. We characterize those irrationa...

Hasse–Schmidt derivations, divided powers and differential smoothness

Let k be a commutative ring, A a commutative k-algebra and D the filtered ring of k-linear differential operators of A. We prove that: (1) The graded ring grD admits a canonical embedding θ into the graded dual of the symmetric algebra of the module ΩA/k of differentials of A over k, which has a canonical divided power structure. (2) There is a canonical morphism θ from the divided power algebr...